Many atomic nuclei possess a magnetic moment. Nuclear magnetic resonance (NMR) is a phenomenon exhibited by this select group of atomic nuclei (termed "NMR active" nuclei), which results from the interaction of the nuclei with an applied, external magnetic field.
The magnetic properties of a nucleus are conveniently discussed in terms of two quantities: the magnetogyric ratio (denoted by the symbol .gamma.); and the nuclear spin (denoted by the symbol I). When an NMR active nucleus is placed in a magnetic field, its nuclear magnetic energy levels are split into (2I+1) non-degenerate energy levels, and these levels are separated from each other by a characteristic energy that is directly proportional to the strength of the applied magnetic field. This phenomenon is called "Zeeman" splitting and the characteristic energy is equal to .gamma.hH.sub.o /2.pi., where h is Plank's constant and H.sub.o is the strength of the magnetic field. The frequency corresponding to the energy of the Zeeman splitting (.omega..sub.o =.gamma.H.sub.o) is called the "Larmor frequency" or "resonance" frequency. Typical NMR active nuclei include .sup.1 H (protons), .sup.13 C, .sup.19 F, and .sup.31 P nuclei. For these four nuclei, the nuclear spin I=1/2, and, accordingly, each nucleus has two nuclear magnetic energy levels.
When a bulk material sample containing NMR active nuclei is placed within a magnetic field, the nuclear spins distribute themselves amongst the nuclear magnetic energy levels in a known manner in accordance with Boltzmann's statics. This distribution results in a population imbalance between the energy levels and a net nuclear magnetization. It is this net nuclear magnetization that is studied by NMR techniques.
At equilibrium, the net nuclear magnetization is aligned with the external magnetic field and is time-independent. A second magnetic field perpendicular to the first magnetic field and rotating at, or near, the Larmor frequency can also be applied to the nuclei and this second field disturbs the equilibrium and induces a coherent motion (a "nutation") of the net nuclear magnetization. Since, at conventional magnetic field strengths, the Larmor frequency of typical NMR active nuclei is in the megahertz range, this second field is called a "radio-frequency field" (RF field). The effect of the RF field is to rotate the spin magnetization about the direction of the applied RF field. The time duration of the applied RF field determines the angle through which the spin magnetization nutates and, by convention, an RF pulse of sufficient length to nutate the nuclear magnetization through an angle of 90.degree. or .pi./2 radians, is called a ".pi./2 pulse".
A .pi./2 pulse applied at a frequency near the resonance frequency will rotate a spin magnetization that was aligned along the external magnetic field direction in equilibrium into a plane perpendicular to the external magnetic field. The component of the net magnetization that is transverse to the external magnetic field then precesses about the external magnetic field at the Larmor frequency. This precession can be detected with a resonant coil located with respect to the sample such that the precessing magnetization induces a voltage across the coil. Frequently, the "transmitter" coil employed to apply the RF field to the sample and cause the spin magnetization to nutate and the "receiver" coil employed to detect the resulting precessing magnetization are one and the same coil. This coil is generally part of an NMR probe.
In addition to processing at the Larmor frequency, the magnetization induced by the applied RF field changes and reverts to the equilibrium condition over time as determined by two relaxation processes: (1) dephasing within the transverse plane ("spin-spin relaxation") with an associated relaxation time, T.sub.2, and (2) a return to the equilibrium population of the nuclear magnetic energy levels ("spin-lattice relaxation") with an associated relaxation time, T.sub.1.
In order to use the NMR phenomenon to obtain an image of a sample, a magnetic field is applied to the sample, along with a magnetic field gradient which depends on physical position so that the field strength at different sample locations differs. When a field gradient is introduced, as previously mentioned, since the Larmor frequency for a particular nuclear type is proportional to the applied field, the Larmor frequencies of the same nuclear type will vary across the sample and the frequency variance will depend on physical position. By suitably shaping the applied magnetic field and processing the resulting NMR signals for a single nuclear type, a nuclear spin density image of the sample can be developed.
When an external magnetic field is applied to a nuclei in a chemical sample, the nuclear magnetic moments of the nuclei each experience a magnetic field that is reduced from the applied field due to a screening effect from the surrounding electron cloud. This screening results in a slight shift of the Larmor frequency for each nucleus (called the "chemical shift" since the size and symmetry of the shielding is dependent on the chemical composition of the sample).
In addition to the applied external magnetic field, each nucleus is also subject to local magnetic fields such as those generated by other nuclear and electron magnetic moments associated with nuclei and electrons located nearby. Interaction between these magnetic moments are called "couplings", and one important example of such couplings is the "dipolar" coupling. When the couplings are between nuclei of like kind, they are called "homo-nuclear couplings". In solids, the NMR spectra of spin=1/2 nuclei are often dominated by dipolar couplings, and in particular by dipolar couplings with adjacent protons. These interactions affect imaging by broadening the natural resonance linewidth and thereby reducing the image resolution.
In order to reduce the effect of such couplings, a class of experiments employs multiple-pulse coherent averaging to continuously modulate the internal Hamiltonians such that, in an interaction frame, selected Hamiltonians are scaled. A sub-class of such experiments is designed to reduce the effects of homonuclear dipolar couplings by averaging the dipolar Hamiltonian to zero over a selected time period in this interaction frame. The most widely used group of these latter experiments consists of long trains of RF pulses applied in quadrature. Data is sampled between groups of pulses.
Multiple-pulse coherent averaging requires that the spin Hamiltonian be toggled through a series of predetermined states, the average of which has the desired property that the dipolar interaction appears to vanish. If an additional requirement is satisfied that the final Hamiltonian state of the series is equivalent to the first Hamiltonian state of the series, the process can be repeated and the temporal response of the sample can be mapped out successively, point-by-point.
If an experiment is performed as described above with the final state exactly equaling the initial state, the result is particularly uninteresting since the result is the same as if nothing had happened to the nuclei between observations. In practice, in order to produce a more interesting result, an interaction of interest, such as a gradient interaction, is allowed to rotate the spin system slightly between successive observations. Although the undesirable interactions, such as dipolar couplings are almost always much larger than the interactions of interest, in some cases the experiment can be cleverly designed so that this slight rotation will have a minimal influence on the averaging of the undesirable interactions. More often though, the effective rotation of the desired interaction does interfere with the coherent averaging process and must be kept within some limits to maintain effective averaging.
For solid-state NMR imaging, in which the dipolar coupling effect is especially strong, a particularly appealing multiple-pulse sequence is a "time-suspension" cycle that aims at averaging all internal Hamiltonians to zero. Therefore, the observed evolution results solely from external interactions, such as the gradient interaction, and the resonance line width is reduced as much as possible. All else being equal, the more success that a technique has in narrowing the resonance linewidth, the better the quality of the image that can be obtained.
For abundant spin=1/2 nuclei in solids, the most obvious undesirable interaction to reduce is the homo-nuclear dipolar interaction, since this is the dominant interaction. The weak coupling of the nuclear magnetic moment to the lattice, the long relaxation times, and the ability to perturb the spin state virtually instantaneously via RF pulses allows the nuclear spin dynamics to be strongly modulated, and it is possible to employ this modulation to periodically refocus selected interactions. In view of this, prior art approaches for line narrowing generally are derived from average Hamiltonian theory. In accordance with this theory, the stationary laboratory reference coordinate system or frame can be mathematically transformed into another coordinate frame which simplifies the observation of an interaction. For example, the nuclear spin dynamics can be viewed in an interaction frame in which the spin dynamics modulation appears as a time-dependent Hamiltonian. In this frame, under a combination of periodic and cyclic conditions, the long-time behavior of the spin system is the time average of the associated interaction frame Hamiltonian over the repeated short cycles.
More particularly, the general approach is outlined as follows. A time-dependent RF field is defined in accordance with the following equation: EQU RF(t)=A.sub.x (t)I.sub.x +A.sub.y (t)I.sub.y ( 1)
where A.sub.x and A.sub.y are time-dependent RF field amplitudes and I.sub.x and I.sub.y are the x and y components of the field vector. If the RF field is periodic, that is, it meets the constraint defined by the following equation: EQU RF(t)=RF(t+T.sub.c) (2)
and cyclic, in that it meets the requirements of the following equation: ##EQU1## then the long term behavior of the spin dynamics will follow the average of the spin dynamics over a single cycle.
For convenience, an interaction frame is normally selected that moves with the spins in response to the RF pulses. This interaction frame is the simplest to deal with since the RF interaction is large, under experimental control and ideally uniform across the sample (that is, all spin packets experience the same interaction frame). The zero-order averaged Hamiltonians that will dictate the spin dynamics are therefore, ##EQU2## for the homo-nuclear dipolar interactions, ##EQU3## for the inhomogeneous interactions, and ##EQU4## for the gradient interaction.
Prior art pulse sequences were designed so that the dipolar and inhomogeneous average Hamiltonians (Equations (4) and (5)) became zero, but the gradient average Hamiltonian (defined by equation (6)) was non-zero. These conditions allowed averaging of the dipolar and inhomogeneous interactions, while leaving the gradient interaction to drive the evolution of the spin dynamics. However, it was found that even when the conditions of equations (4)-(6) were met as described, high quality images were not obtained. This happened because the gradient evolution is as strong an interaction as the RF pulse interaction and, consequently, could not be neglected in defining the interaction frame.
When the gradient interaction is taken into account in defining the interaction frame, the correct zero-order average Hamiltonian appears as defined by the following: ##EQU5## instead of the simplified version in Equation (4). The gradient appears in the interaction frame transformation as expected, but the transformation, and hence the average dipolar Hamiltonian, also depends on the spin packet's spatial location. This dependency introduces a spatial heterogeneity, which, in turn, has led to images with resolutions that vary with image location (most often yielding an image with high resolution at the center and decreasing resolution towards the edges). The same effect is encountered in NMR spectroscopy where the spectral resolution often varies with the frequency offset from resonance.
It has been found that it is possible to remove the normal spatial dependence from the interaction frame which is embodied in Equation (7). In particular, in one prior art experiment, this is done by (1) arranging the RF pulse sequences and the gradient pulse sequences so that the RF fields and the gradient fields do not overlap in time, and (2) selecting RF pulse sequences so that the sub-cycles of RF modulation between gradient pulses are either cyclic or anti-cyclic (that is have propagators that equal +1 or -1).
The effect of these conditions is to allow the RF propagator and the gradient propagator to commute in such a manner that the gradient and RF interactions become decoupled as explained in detail below. However, this decoupling occurs only to the zero-order approximation. The higher order terms in the Magnus expansion are not negligible and still cause coupling between the line-narrowing and gradient interactions which results in a variation of the resolution over an image, albeit that this latter variation is reduced from the variation in the non-decoupled case.
Accordingly, it is an object of the present invention to provide a method for operating a solid-state NMR imaging system so that the resulting images have more uniform resolution over the entire image.
It is another object of the present invention to provide a method for operating a solid-state NMR imaging system in which line-narrowing can be carried out without introducing an inhomogeneity in the overall spatial resolution.
It is another object of the present invention to provide a method for operating a solid-state NMR imaging system in which the spatial dependency of the average dipolar Hamiltonian which results from line-narrowing is eliminated to a second order approximation.
It is still another object of the present invention to provide a method for operating a solid-state NMR imaging system in which the spatial resolution is uniform and which method can be used with a variety of conventional line-narrowing RF pulse sequences.